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870, 12, 3, 2

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  • Mark Underwood
    A little tidbit: The eight possible (positive) additive combinations of the four numbers 870, 12, 3, and 2 yield the eight consecutive primes 853, 857, 859,
    Message 1 of 8 , Nov 14, 2008
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      A little tidbit: The eight possible (positive) additive combinations of the four numbers
      870, 12, 3, and 2
      yield the eight consecutive primes
      853, 857, 859, 863, 877, 883 and 887.

      I haven't checked throughly, but this might be the lowest such run of eight consecutive
      primes produced in this manner.

      Mark
    • Mark Underwood
      ... Sorry, forgot 881. Now for some silliness: Consider the year 1995, and the numbers 3,7 and 12. The eight possible positive additive combinations of those
      Message 2 of 8 , Nov 14, 2008
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        --- In primenumbers@yahoogroups.com, "Mark Underwood" <mark.underwood@...>
        wrote:
        >
        > A little tidbit: The eight possible (positive) additive combinations of the four numbers
        > 870, 12, 3, and 2
        > yield the eight consecutive primes
        > 853, 857, 859, 863, 877, 883 and 887.
        >
        > I haven't checked throughly, but this might be the lowest such run of eight consecutive
        > primes produced in this manner.
        >
        > Mark
        >

        Sorry, forgot 881.

        Now for some silliness: Consider the year 1995, and the numbers 3,7 and 12. The eight
        possible positive additive combinations of those numbers yield all prime years:

        1973, 1979, 1987, 1993, 1997, 2003, 2011 and 2017.

        Alas they are not quite consecutive. 1999 is missing.

        Mark
      • Jens Kruse Andersen
        ... According to my computations, if the additive combinations of 5 numbers must give 16 distinct primes then the smallest possible difference between the
        Message 3 of 8 , Nov 15, 2008
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          Mark Underwood wrote:
          > A little tidbit: The eight possible (positive) additive combinations of
          > the four numbers
          > 870, 12, 3, and 2
          > yield the eight consecutive primes
          > 853, 857, 859, 863, 877, 881 (added in followup), 883 and 887.

          According to my computations, if the additive combinations of
          5 numbers must give 16 distinct primes then the smallest possible
          difference between the smallest and largest prime is 164.
          This can be reached for x, 42, 18, 12, 10.
          The primes are x+/-n, for n = 2, 22, 26, 38, 46, 58, 62, 82.
          The first occurrence is at x = 9894203506406653455.
          The primes are not consecutive. x+8, x+68 and x+74 are also prime.

          --
          Jens Kruse Andersen
        • Jens Kruse Andersen
          ... Correction: The above difference is only minimal if the 4 small numbers are required to be even. Otherwise the smallest admissible difference is 82,
          Message 4 of 8 , Nov 15, 2008
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            I wrote:
            > According to my computations, if the additive combinations of
            > 5 numbers must give 16 distinct primes then the smallest possible
            > difference between the smallest and largest prime is 164.
            > This can be reached for x, 42, 18, 12, 10.

            Correction: The above difference is only minimal if the 4 small
            numbers are required to be even.
            Otherwise the smallest admissible difference is 82, reached by
            halving the above numbers to y, 21, 9, 6, 5.

            --
            Jens Kruse Andersen
          • Mark Underwood
            ... Interesting. Jens I honestly don t know how it is you can check all the way up to that huge number! But I like some mystery. :) All I know is that x must
            Message 5 of 8 , Nov 15, 2008
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              --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
              <jens.k.a@...> wrote:
              >

              > According to my computations, if the additive combinations of
              > 5 numbers must give 16 distinct primes then the smallest possible
              > difference between the smallest and largest prime is 164.
              > This can be reached for x, 42, 18, 12, 10.

              >****
              >Jens added correction:
              >The above difference is only minimal if the 4 small
              >numbers are required to be even.
              >Otherwise the smallest admissible difference is 82, reached by
              >halving the above numbers to y, 21, 9, 6, 5.
              >****

              > The primes are x+/-n, for n = 2, 22, 26, 38, 46, 58, 62, 82.
              > The first occurrence is at x = 9894203506406653455.
              > The primes are not consecutive. x+8, x+68 and x+74 are also prime.
              >

              Interesting. Jens I honestly don't know how it is you can check all
              the way up to that huge number! But I like some mystery. :) All I
              know is that x must be a multiple of 15 given your even n's, and a
              multiple of 30 given your odd n's. (Oh yes, and not having a factor of
              11, 13, 19, 23, 29,31 and 41.)

              It looks like you have found: the first occurrence of the tightest
              possible bunching of 16 primes derived from the addititive
              combinations of 5 numbers.

              When the 16 numbers become consecutive then I suppose we are merging
              into the area of admissible prime constellations. I'm just guessing
              (and hoping) that the first occurrence of 16 consecutive primes
              generated by the additive combinations of 5 numbers will occur well
              before your huge x. But it won't be the tightest possible bunching, as
              you have found.

              Since this morning my old faithful Windows 98 (1st edition, heh!)
              computer has been chugging away at the additive combinations of 6
              numbers, numbers up to 300. It should be done by tomorrow. I think it
              was last year when I found one case of 32 distinct primes generated by
              the additive combination of six numbers. I only looked at numbers up
              to 150. David Broadhurst was the first to find six numbers which in
              additive combination generated all primes, and we were surprised it
              was achieved so quickly. I think the highest of the six numbers was
              105, yielding 24 (or 28?) distinct primes.

              Anyways I'll report tomorrow or Monday on what I've found. I've also
              found a couple of things which seem puzzling but I need to check it
              out more carefully to make sure I'm not overlooking the obvious.


              Mark
            • Jens Kruse Andersen
              ... I used an unpublished tuple search program which has found hundreds of tuple patterns since 2003. Thanks for giving me an opportunity to use it again. A
              Message 6 of 8 , Nov 15, 2008
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                Mark Underwood wrote:
                > <jens.k.a@...> wrote:
                >> The primes are x+/-n, for n = 2, 22, 26, 38, 46, 58, 62, 82.
                >> The first occurrence is at x = 9894203506406653455.
                >
                > Interesting. Jens I honestly don't know how it is you can check all
                > the way up to that huge number!

                I used an unpublished tuple search program which has found hundreds
                of tuple patterns since 2003.
                Thanks for giving me an opportunity to use it again.
                A few parameters just have to be changed for each pattern and then
                the program works out admissible values modulo each small prime.

                > It looks like you have found: the first occurrence of the tightest
                > possible bunching of 16 primes derived from the addititive
                > combinations of 5 numbers.

                Due to an initial mistake, I only found the first occurrence of
                the bunching which is the tightest possible when the 4 smallest
                numbers are all even. If some of them are allowed to be odd
                (and I see no reason to disallow that) then I'm searching for
                the first occurrence but may stop before succeeding.
                I chose to search a specific tight bunching because I already
                had a suitable program for a fixed pattern, and a tight bunching
                is likely to have consecutive primes.

                32 distinct primes generated by the additive combination of six
                numbers seems relatively easy. There are large solutions based
                on other published results. For example, any two non-overlapping
                AP16 with the same common difference will give a solution.
                http://hjem.get2net.dk/jka/math/aprecords.htm has more than needed:
                AP19 and AP20 by Jaroslaw Wroblewski.
                AP19: 254215977184797362303 + 53#*n, n = 0..18
                AP20: 178284683588844176017 + 53#*n, n = 0..19
                They have respectively 4 and 5 AP16 as subsequences so this
                gives 4*5 = 20 AP16-based solutions in total.
                The solution with the last AP16 in both cases gives 32 21-digit primes for
                574731073635911261190, 21671067559381570778, 4*53#, 2*53#, 53#, 53#/2.
                Here 574731073635911261190 +/- 21671067559381570778 is
                in the middle of the two chosen AP16.

                In 2004 Gennady Gusev and I published the first 3 AP17 with
                common difference 17#:
                http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0410&L=nmbrthry&F=&S=&P=2620
                In total they give 2*2*3 = 12 pairs of AP16's.
                By the way, the same tuple program as above was used.

                --
                Jens Kruse Andersen
              • Jens Kruse Andersen
                I returned to this after completing some other tasks. 320572022166380880 +/- 30 +/- 9 +/- 5 +/- 3 gives 16 distinct consecutive primes. They can also be
                Message 7 of 8 , Dec 1, 2008
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                  I returned to this after completing some other tasks.

                  320572022166380880 +/- 30 +/- 9 +/- 5 +/- 3 gives 16 distinct
                  consecutive primes. They can also be written:
                  320572022166380880 +/- n, for n = 13, 19, 23, 29, 31, 37, 41, 47.

                  It is the shared second tightest admissible pattern for
                  16 distinct primes produced by the additive combinations of
                  5 numbers where the largest is always added.
                  The 37 tightest patterns were searched to some limit.
                  4 of them had a case with 16 primes but only one case had
                  consecutive primes.
                  The 5 numbers, the difference between the 1st and 16th prime,
                  and the number of other primes between them are:
                  {320572022166380880, 30, 9, 5, 3}, difference 94, 0 other.
                  {87291414128856315, 33, 18, 12, 5}, difference 136, 2 other.
                  {57312341532495501, 24, 21, 15, 10}, difference 140, 1 other.
                  {82911614607, 45, 15, 7, 3}, difference 140, 1 other.

                  The last case was a surprise at only 11 digits.
                  It is the only occurrence of that pattern below 10^18.

                  --
                  Jens Kruse Andersen
                • Mark Underwood
                  ... Incredible find! And of course the prime tuple or constellation is symmetrical to boot. ... Hey, some of us rely on surprises that low , hehe. Mark
                  Message 8 of 8 , Dec 1, 2008
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                    --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
                    >
                    > I returned to this after completing some other tasks.
                    >
                    > 320572022166380880 +/- 30 +/- 9 +/- 5 +/- 3 gives 16 distinct
                    > consecutive primes. They can also be written:
                    > 320572022166380880 +/- n, for n = 13, 19, 23, 29, 31, 37, 41, 47.
                    >
                    > It is the shared second tightest admissible pattern for
                    > 16 distinct primes produced by the additive combinations of
                    > 5 numbers where the largest is always added.

                    Incredible find! And of course the prime tuple or constellation is symmetrical to boot.


                    > The 37 tightest patterns were searched to some limit.
                    > 4 of them had a case with 16 primes but only one case had
                    > consecutive primes.
                    > The 5 numbers, the difference between the 1st and 16th prime,
                    > and the number of other primes between them are:
                    > {320572022166380880, 30, 9, 5, 3}, difference 94, 0 other.
                    > {87291414128856315, 33, 18, 12, 5}, difference 136, 2 other.
                    > {57312341532495501, 24, 21, 15, 10}, difference 140, 1 other.
                    > {82911614607, 45, 15, 7, 3}, difference 140, 1 other.
                    >
                    > The last case was a surprise at only 11 digits.
                    > It is the only occurrence of that pattern below 10^18.

                    Hey, some of us rely on surprises that 'low', hehe.

                    Mark
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